The number of values of x such that `x, [x] and {x}` are in arithmetic progression is equal to (where `[.]` denotes the greatest integer function and `{.}` denotes the fractional pat function)

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem of finding the number of values of \( x \) such that \( x \), \( [x] \), and \( \{x\} \) are in arithmetic progression, we will follow these steps:

### Step-by-Step Solution

1. **Understanding the Functions**:
- Let \( x \) be expressed as \( x = [x] + \{x\} \), where \( [x] \) is the greatest integer less than or equal to \( x \) and \( \{x\} = x - [x] \) is the fractional part of \( x \).

2. **Setting Up the Arithmetic Progression**:
- For \( x \), \( [x] \), and \( \{x\} \) to be in arithmetic progression, we must have:
\[ 2[x] = x + \{x\} \]

3. **Substituting the Expression for \( x \)**:
- Substitute \( x = [x] + \{x\} \) into the equation:
\[ 2[x] = ([x] + \{x\}) + \{x\} \]
- This simplifies to:
\[ 2[x] = [x] + 2\{x\} \]

4. **Rearranging the Equation**:
- Rearranging gives us:
\[ [x] = 2\{x\} \]

5. **Understanding the Range of \( \{x\} \)**:
- The fractional part \( \{x\} \) lies in the interval \( [0, 1) \). Therefore, \( 2\{x\} \) will lie in the interval \( [0, 2) \).

6. **Finding Possible Values for \( [x] \)**:
- Since \( [x] \) is an integer, we have:
\[ 0 \leq [x] < 2 \]
- This means \( [x] \) can take the integer values \( 0 \) or \( 1 \).

7. **Calculating Corresponding Values of \( x \)**:
- **Case 1**: If \( [x] = 0 \):
\[ 0 = 2\{x\} \implies \{x\} = 0 \implies x = [x] + \{x\} = 0 + 0 = 0 \]
- **Case 2**: If \( [x] = 1 \):
\[ 1 = 2\{x\} \implies \{x\} = \frac{1}{2} \implies x = [x] + \{x\} = 1 + \frac{1}{2} = \frac{3}{2} \]

8. **Conclusion**:
- The values of \( x \) that satisfy the condition are \( 0 \) and \( \frac{3}{2} \).

### Final Answer:
The number of values of \( x \) such that \( x \), \( [x] \), and \( \{x\} \) are in arithmetic progression is **2**.